Fractions constructible with n different-sized squares

A ratio, e.g. 200/117, is represented as a 200×117 rectangle. Operation of the Euclidean algorithm or the jigsaw method gives the continued fraction [1;1,2,2,3,1,3].

On this page, variables are positive integers. Rectangles are drawn only for , but usually in the text, only the proper fractions with are given, to save space (and these lie conveniently between 0 and 1.) But remember, the rectangle represents both and e.g. where appears below, also has the same number and size of constituent squares, it is just rotated (which here = inversion).

1 size

1 square alone is or .
Putting 2, 3, 4… squares next to each other, we have the numbers 2, 3, 4.. or, looking from the direction, , , …
i.e. numbers of the form and
In order of magnitude:
Or, from the direction,

N.B. From here on I will omit all mention of the half of the fractions , except in the general formulas, to save space. For every mentioned, is in the same group.

2 sizes

Using 1 large square and smaller ones along the side, we get
[illustrate]
Using 2 squares and smaller ones along the side,
[illustrate a few]
Using 3 and smaller ones,
[show a few]
i.e. numbers of the form (and ), where .
[show diagram of general case]
In order of magnitude:

[picture of plot of these on number line, , each a short vertical line, maybe pic width 500px]

3 sizes
Working backwards, this group is made up of each of the previous group, with 1 or more squares stuck on to the side.
From the “2 sizes, 1 large square group”: with 3×3 squares added makes: with 4×4 squares added makes: with 5×5 squares added makes: etc.

4 sizes

Functions

Define: the number of Different-sized squares needed to construct . number of iterations of Euclid’s algorithm to find number of terms in the continued fraction.

the total Number of squares needed to construct , regardless of size. sum of the continued fraction terms . total of the numbers used as multipliers in Euclid’s algorithm.

If a/b is irrational, is infinite, e.g. . Although e.g. for = VALUE*#*#*#?!?, but .

[picture of Stern-Brocot tree, , with D() and N() next to each number. width =500px?]

=== state rules for D and N for S-B tree. and for adding fractions generally.

=== make table of D and table of N for a=1…50 and b=1…50 .. or 30, whatever fits on a page.
===could do a table 100×100 with dots in each square representing D(a/b).. or colour coded.. could go higher.
=== what WOULD that look like?! 🙂 or even 500×500, 2 pixel colour in each one… 250×250 maybe better.

=== How many fractions have =1,2,3..? is is Catalan numbers or something?